On the length of the central series (derived series) of a Moufang commutative loop
A loop \((Q,\cdot)\) is a Moufang commutative loop (MCL) if \(a^2\cdot bc=ab\cdot ac\) for all \(a,b,c\in Q\). The associator \([a,b,c]\) is defined by \(ab\cdot c=a[a,b,c]\cdot bc\). The lower (transfinite) central series of an MCL \(Q\) has the form \(Q=Q_0\supseteq Q_1\supseteq\cdots\supseteq Q_i\supseteq Q_{I=1}\supseteq\cdots\), where the normal subloop \(Q_{i+1}\) is generated by all associators \([a,u,v]\) with \(a\in Q_i\) and \(u,v\in Q\). Similarly \(Q=Q^{(0)}\supseteq Q^{(1)}\supseteq\cdots\supseteq Q^{(i)}\supseteq Q^{(i+1)}\supseteq\cdots\) is called the (transfinite) derived series of MCL \(Q\) if \(Q^{(i+1)}\) is the associant (subgroup generated by all associators) of the subgroup \(Q^{(i)}\). The least transfinite \(n\) for which \(Q_n=1\) resp. \(Q^{(n)}=1\) is called the length of the respective series. In this paper, by analogy with group theory, the author proves that there exists an MCL for which the length of the lower central series resp. derived series is equal to a prescribed transfinite number.
scientific article; zbMATH DE number 1257046